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Large deviations for invariant measures of multivalued stochastic differential equations with jumps

Huijie Qiao

TL;DR

This work develops uniform large deviation principles for multivalued stochastic differential equations with jumps, addressing both the state process and its invariant measures. By employing the weak convergence approach and carefully constructing skeleton equations that incorporate the maximal monotone operator A, the authors establish Freidlin-Wentzell and Dembo-Zeitouni uniform LDPs under general structural conditions, including convergence results in the Skorokhod topology. A key contribution is the LDP for invariant measures, characterized by a quasi-potential V(y) and proven via exponential moment estimates and Yosida-approximation techniques. These results extend prior uniform LDPs to settings with both A and jump terms, with implications for metastability and long-time behavior of constrained stochastic systems.

Abstract

This work focuses on multivalued stochastic differential equations with jumps. First, by employing the weak convergence approach, we establish the Freidlin-Wentzell uniform large deviation principle and the Dembo-Zeitouni uniform large deviation principle for these equations. Subsequently, based on these results, we derive both upper and lower bounds for the large deviations of invariant measures associated with the equations.

Large deviations for invariant measures of multivalued stochastic differential equations with jumps

TL;DR

This work develops uniform large deviation principles for multivalued stochastic differential equations with jumps, addressing both the state process and its invariant measures. By employing the weak convergence approach and carefully constructing skeleton equations that incorporate the maximal monotone operator A, the authors establish Freidlin-Wentzell and Dembo-Zeitouni uniform LDPs under general structural conditions, including convergence results in the Skorokhod topology. A key contribution is the LDP for invariant measures, characterized by a quasi-potential V(y) and proven via exponential moment estimates and Yosida-approximation techniques. These results extend prior uniform LDPs to settings with both A and jump terms, with implications for metastability and long-time behavior of constrained stochastic systems.

Abstract

This work focuses on multivalued stochastic differential equations with jumps. First, by employing the weak convergence approach, we establish the Freidlin-Wentzell uniform large deviation principle and the Dembo-Zeitouni uniform large deviation principle for these equations. Subsequently, based on these results, we derive both upper and lower bounds for the large deviations of invariant measures associated with the equations.

Paper Structure

This paper contains 10 sections, 9 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Maximal monotone operators
  5. Uniform LDPs
  6. Main results
  7. Two uniform LDPs for multivalued SDEs with jumps
  8. The Freidlin-Wentzell LDP for invariant measures of multivalued SDEs with jumps
  9. Proof of Theorem \ref{['fwuldpth']} and \ref{['dzuldpth']}
  10. Some estimates for $X^{\varepsilon,\xi^\varepsilon, \zeta^\varepsilon}, X^{\xi, \zeta}$

Key Result

Lemma 2.4

For $X\in D([0,T],\overline{{\mathcal{D}}(A)})$ and $K\in {\mathscr V}_{0}$, the following statements are equivalent:

Theorems & Definitions (23)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Definition 2.7: Freidlin-Wentzell uniform LDP
  • Definition 2.8: Uniform Laplace principle
  • Theorem 2.10
  • Remark 3.1
  • ...and 13 more